In this paper we prove that every totally real algebraic integer \(\lambda \) of degree \(d \ge 2\) occurs as an eigenvalue of some tree of height at most \(d(d+1)/2+3\). In order to prove this, for a given algebraic number \(\alpha \ne 0\), we investigate an additive semigroup that contains zero and is closed under the map \(x \mapsto \alpha /(1-x)\) for \(x \ne 1\). The problem of finding the smallest such semigroup seems to be of independent interest.

中文翻译：

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给定特征值的树高度的上限

在本文中，我们证明每个度为\( d \ge 2\) 的全实代数整数 \(\lambda \)作为某个高度最多为\(d(d+1)/2+3的树的特征值出现\)。为了证明这一点，对于给定的代数数\(\alpha \ne 0\)，我们研究一个包含零且在映射\(x \mapsto \alpha /(1-x)\)下闭合的加法半群对于\(x \ne 1\)。找到最小的这样的半群的问题似乎是具有独立意义的。